If $f: S\to \mathbb{R}$ and $f$ is differentiable of class $C^r$ at each point $x_0$ of $S$, then $f$ may be extended to a $C^r$ function

Question

This problem has already been posted here without an answer: $f$ may be extended to a $C^r$ function $h : A \to R$.

I have thought about this problem but I do not know how I can begin to solve it, I think that the following theorem can be used but I do not know how. Could anyone help me, please? Thank you very much.